T_z constraints of semi-elliptical surface cracks in elastic plates subjected to uniform tension loading

The out-of-plane constraints T_z around the semi-elliptical surface cracks in an elastic plate subjected to uniform tension loading have been investigated through detailed three-dimensional (3D) finite element (FE) analyses. The distributions of T_z are obtained in the vicinity of the crack border with aspect ratios of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. T_z drops from Poisson's ratio at the crack tip to approximate zero beyond certain radial distance in the normal plane of the crack front line, and increases gradually from the free surface to the mid-plane at the same radial distance. By fitting the numerical results, empirical formulae are obtained to describe the 3D distributions of T_z for semi-elliptical surface cracks with a sufficient accuracy in the wide aspect ratio range of 0.2 ≤ a/c ≤ 1.0 except very near the free surface, where T_z is extremely low. T_z, combining with the corresponding K and T or J and Q, can be applied to establish the three-parameter dominated stress field, which can characterize the 3D crack front field completely as an attempt.